Would a HD wallet generated with a 24 words passphrase attached to a 24 word seed phrase that is know to an attacker be as hard to brute force as a HD wallet generated using only 24 words and no passphrase?
That's an interesting question. In general, both a 24-word passphrase and a 24-word seed phrase of a hierarchical deterministic (HD) wallet can provide an extremely high level of security. If they are carefully selected and prepared, it becomes practically impossible to brute force them.
However, since the words in the 24-word seed phrase are randomly selected from a 2048-word dictionary, the number of possible combinations can be easily calculated: that's 24!
When considering a 24-word passphrase, things get a lot more complicated. Unlike a seed phrase for a HD wallet, a passphrase can be constructed using any words, including made-up words, derivatives, and abbreviations. For example, according to some research from Harvard University and Google, the English language alone has more than a million words, and the average native English speaker can use between 20,000 and 30,000 words on average. Additionally, unlike a seed phrase, a passphrase can contain the same word an unlimited number of times. Due to the virtually limitless word choices and the potential for creating complex phrases, calculating the exact number of possible combinations becomes nearly impossible. So to conclude, the number of combinations for a 24-word passphrase constructed from such a vast word pool would far exceed the already astronomically large number of combinations for a 24-word seed phrase from a fixed 2048-word dictionary.